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L10f05

# L10f05 - CO350 Linear Programming Chapter 4 Introduction to...

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Unformatted text preview: CO350 Linear Programming Chapter 4: Introduction to Duality 25th May 2005 Chapter 4: Introduction to Duality 1 Economic Interpretation of Duality (Section 4.6, Pg 51) General Production Problem A factory makes n products from m resources. Each unit of product j requires a ij units of resource i and makes a profit of c j dollars. Each day, the factory has b j units of resource i available. Mathematical Model maximize P n j =1 c j x j subject to P n j =1 a ij x j ≤ b i ( i = 1 , 2 ,...,m ) x j ≥ 0 ( j = 1 , 2 ,...,n ) Suppose the resource constraints are not hard; i.e., we can either buy or sell each resources at certain fixed price. What is the fair price for each resource? Fair Price: There is no advantage to either buy or sell small amount the resource. Fair prices are also called shadow prices . Chapter 4: Introduction to Duality 2 Example (Not in notes) maximize 2 x 1 + 3 x 2 subject to 2 x 1 + x 2 ≤ 10 (Electricity) x 1 + x 2 ≤ 6 (Oranges)- x 1 + x 2 ≤ 4 (Water) x 1 , x 2 ≥- x 1 6 x 2 A A A A A A A A A A A A Electricity @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Oranges Water Feasible Region Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q...
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L10f05 - CO350 Linear Programming Chapter 4 Introduction to...

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